3.2 Channel Estimation and Equalization
3.2.1 Training based Channel Estimation and Equalization
In order to estimate the source signals, training signals are first transmitted for channel estimation. Then, equalization is performed using the CSI estimated by the training signals.
In this section, three widely used channel estimation schemes are discussed: LS based channel estimation, MMSE based channel estimation [12] [63] and channel interpolation [64] methods.
LS based Channel Estimation The LS based method has low complexity, and thus has been widely used for channel estimation. Defineh= [ho, . . . , hL−1, hL, . . . , hN−1]T as the channel vector, of which elements are assumed to be Gaussian variables and independent to each other. If there are a total number of L channel paths, thenhL =
. . .=hN−1 = 0. The channel response energy is normalized to unity asPNl=0−1E{h2l}= 1. In SISO systems, the received signal vector y(i) = [y(0, i), y(1, i), . . . , y(N −1, i)]T in thei-th block can be written as
y(i) =X(i)H+z(i), (3.10)
where X(i) = diag{[x(0, i), x(1, i), . . . , x(N −1, i)]T} is the N ×N diagonal training matrix, H = √NFh is the channel frequency response vector on N subcarriers, and
z(i) is the noise vector.
Assuming a total number of Ns blocks as training, the LS based channel estimate
ˆ HLS can be given by ˆ HLS= 1 Ns Ns−1 X i=0 X−1(i)y(i). (3.11)
Substituting Equations (3.10) into (3.11) yields ˆ HLS= 1 Ns Ns−1 X i=0 H+X−1(i)z(i). (3.12)
The termX−1(i)z(i) may be subject to noise enhancement, especially when the channel is in a deep null.
For MIMO OFDM systems, channel estimation can be decoupled into a number of independent SISO OFDM channel estimations, if the training symbols of transmit an-
tennas are orthogonal to each other [65]. It means that a different subset of subcarriers is used by each transmit antenna for the training symbols transmission.
MMSE based Channel Estimation MMSE usually outperforms LS in channel estimation, as it can suppress the noise enhancement for known channel characteristics [66]. The MMSE based channel estimate ˆHMMSE is performed by minimizing the following MSE
min E{||HˆMMSE−H||2}. (3.13)
By using the LS based channel estimate, the MMSE based channel estimation method can be given by [67]
ˆ
HMMSE =RHH(RHH+σz2IN)−1HˆLS, (3.14)
whereRHH= E{HHH} is the auto-correlation of channel frequency response.
Channel Interpolation Channel interpolation [64] can be employed to refine the channel estimate performed by the LS based method. The correlation between adjacent subcarriers is used to correct some incorrect channel estimates for a few subcarriers. By using the LS based channel estimation ˆHLS in Equation (3.12), the time-domain channel estimation ˜h is given as [64]
˜
h= √1 NF
+
N×LHˆLS, (3.15)
where (·)+ denotes the pseudo-inverse, FN×L is the N ×L DFT matrix, with entry
(a, b) given by FN×L(a, b) = √1Ne −j2πab
N (a= 0,1, . . . , N −1;b = 0,1, . . . , L−1). The
channel information for all subcarriers is used so that ˜his not influenced by a few errors on some subcarriers. After inserting (N −L) zeros at the end of ˜h, the N ×1 vector ˆ
h is written as ˆh = [˜h,0, . . . ,0]T. The refined channel estimate ˆH
CI in the frequency domain can be given as
ˆ
HCI=
√
By using the estimate of the CSI, equalization is performed to recover the transmit- ted signal on each subcarrier in OFDM based wireless communication systems. In this subsection, a number of equalization schemes are presented here: ZF and MMSE based equalization, Vertical Bell Laboratories Layered Space-Time (V-BLAST) based equalization and ML based equalization approaches.
ZF based Equalization In order to solve the problem of the CCI and ISI in OFDM based wireless communication systems, the ZF based equalizer applies the inverse of the channel frequency response to the received signal in the frequency domain. Due to the simplicity, it has been widely used in wireless communications systems. However, the noise power might be enhanced after the process, particularly for the subcarriers in the deep fading. When such a consequence arises, the received signal energy may be weak at some frequencies. According to Equation (2.39), the ZF based equalization method is performed on then-th subcarrier as
ˆs(n, i) =GZF(n)y(n, i), (3.17)
whereGZF(n) is the ZF equalizer on then-th subcarrier, given by
GZF(n) = [H(n)HH(n)]−1HH(n). (3.18)
MMSE based Equalization Similarly, the MMSE based equalization scheme out- performs that of ZF in terms of noise power reduction. According to Equation (2.39), the MMSE based equalization method is to optimize the MSE as
GMMSE(n) = arg min G(n)
E
i{||ˆs(n, i)−s(n, i)||
2}. (3.19)
Minimizing Equation (3.19) with respect toG(n) results in the MMSE equalizer as
The source symbols have a unit variance and are spatially uncorrelated as Rss =
E
i{s(n, i)s
H(n, i)} = IM, and the noise is spatially uncorrelated with variance σ2
z as Rnn = E
i{z(n, i)z
H(n, i)}=σ2
zIM.
V-BLAST based Equalization The V-BLAST based equalization [2] [68] method requires a ZF or MMSE equalizer. However, it outperforms both ZF and MMSE in terms of interference cancellation by using ordering to detect the substream with high post-detection SNR. Therefore, the V-BLAST based equalization method can provide a better BER performance than ZF or MMSE based equalization approach. Assuming a number of K transmit antennas, the V-BLAST procedure is summarized below. Forg= 1 :K
1. The MMSE or ZF based equalizer is obtained from the channel.
2. Thek-th substream (k= 0,1, . . . , K−1) in the received signals, corresponding to thekg-th column of the channel with the highest post-detection SNR (or thekg-th
row of the equalizer with the lowest minimum norm square value), is selected in theg-th detection.
3. The hard estimation of thent-th substream is obtained from the equalized signals.
4. The estimated signal on thent-th stream is canceled from the received signals.
5. Thent-th column of the channel or thent-th row of the equalizer is set to null.
End.
Equalization and interference cancellation repeat a number of times until all substreams are detected.
ML based Equalization The ML based equalization method [66] is powerful, which applies the LS based channel estimate, while searching over the possible transmitted
systems can be written as ˆ s(n, i) = arg min ˜ s(n,i) {|| x(n, i)−H(n)˜s(n, i)||2F}, (3.21)
wherex(n, i) and ˜s(n, i) are the received signal vector and the trial transmitted signal vector, respectively, H(n) is the channel frequency response matrix on the n-th sub- carrier between all K transmit antennas and all M receive antennas. The ML based equalization method requires a number of searches, resulting in very high complexity.